Polyform Base
A base is the simplest shape/shapes that are combined together to create polyforms. Base types Regular Polygons Polyiamonds - based on regular triangles, or the triangular tiling Polyominoes - based on regular squares, or the square tiling : Polykings - a variation of the polyominoes that allow vertex connectivity Polypents - based on the regular pentagons. Since they do not tile the plane, there is no grid. Polyhexes - based on regular hexagons, or the hexagonal tiling All higher order polygons are possible to use as a base, but are not explored here at this time. Isohedral Grids Isohedral tilings are face regular tlings, and duals of the vertex regular graphs. Polycairos - based on the Cairo tiling Polyprisma - based on the isohedral Prismatic Pentagonal tiling. Polyrhombilles - based on the isohedral Rhombille tiling of the 60o rombus. Polyflorets - based on the isohedral Floret tiling. Polytriakis - based on the isohedral Triakis tiling of the 30o-30o-120o triangle. Polytetrakis - based on the isohedral Tetrakis tiling of the 45o-45o-90o triangle. Polykites - based on the isohedral 60o-90o-120o-90o quadralateral. Polyrhombillekis - based on the isohedral Rhombillekis tiling of the 30o-60o-90o triangle. Freeform Shapes Polytans - based on the freeform version of the 45o-45o-90o triangle. Polypons - based on the freeform version of the 30o-30o-120o triangle. Polydrafters - based on the freeform version of the 30o-60o-90o triangle. Semiregular Grids Semiregular tilings are vertex regular uniform tilings of the plane, but are not face regular. This means they consist of different shapes, such as the truncated square tiling, which is composed of both squares and octogons. It is possible to use each face as a seperate unit for creating polyforms, but this results in a large range of sizes for a given number of shapes, for example, again with the trucated square tiling, 5-forms could consist of five octogons, down to one octogon and four squares. This makes a lot of topics, such as Isohedral Plane Tiling or Minimal Shape Covering, difficult or meaningless for most forms because of ratio issues. A better way is to consider the single form to consist of a set of shapes in the base ratio for the tiling. Truncated square grid - consisting of squares and octogons in a ratio of 1:1 Snub square grid - consisting of triangles and squares in a ratio of 2:1 Trihexagonal grid - consisting of triangles and hexagons in a ratio of 2:1 Truncated hexagonal grid - consiating of triangles and duodecagons in a ratio of 2:1 Rhombitrihexagonal grid - consisting of triangles, squares, and hexagons in a ratio of 2:3:1 Truncated trihexagonal grid - consisting of squares, hexagons, and duodecagons in a ratio of 3:2:1 Snub hexagonal grid - consisting of two types of triangles, and hexagons in a ratio of 6:2:1 Elongated triangular grid - consisting of triangles and squares in a ratio of 2:1 Other Polygons All triangles and quadrilaterals can tile the plane in basic ways, beyond the special isohedral cases above. Using different types will create different polyforms. Triangles There are three basic triangles shapes, equilateral, isosceles, and scalene. The angles do not matter too much beyond aesthetics and clarity. Equilateral triangles create the polyiamonds. There are four other possible tilings (I do not know of any special names for these tiling, so they don't have links, yet) Isosceles, 180o around each edge Scalene, 180o around each edge Scalene, 180o around two edges, and mirror image along the third. Right angle scalene, 180o around the hypotenuse, mirror image along the other two sides. Which edge is mirrored along doesn't matter, since each tiling would be equivalent, as would the isosceles version. Quadrilaterals All quadrilaterals tile the plane. The general method is for edges connect to the same edge rotated 180o. There are some special cases covered above. The square is the basis of the polyominoes, which are very symetrical, but as the symetries are broken, new types of polyform emerge. Polyedges Polyedges are based in unit line segments that are connected at their ends. The most common are six, four, and three edges spaced evenly around a vertex which prevent edges from overlapping. There are two other Isotoxal tilings that can be used as a basis for polyedges, the Rhombille tiling, and the Trihexagonal tiling. Other vertex counts are possible, but tend to be very chaotic as more edges are added. One that is works well is five edges. "Dented" polygons A dented polygon is a regular polygon with a portion of it's edges inside out, so that it resembles a crescent moon. Basically, for a non-diameter chord between two non adjacent vertices, the smaller arc is replaced by its mirror image in the chord. So for an n-gon, there is a dented polygon for each dent size, d, of 2 \le d < {n \over 2} . Construction rules The rules for combining the shapes can very, and will produce different results depending on which rules are use. The basic rules used most often are: #Two shapes are joined along a single edge, of the same length. #Shapes may not overlap. #A polyform has to be edge connected. #A polyform and it's mirror image are not distinct. Variant rules Grid vs Freeform One way of looking at polyforms is by building them up from single basic shapes. another way is cutting groups of the basic shape out of a regular tiling of the plane by the shape. For the regular triangle, square, and hexagon, that can tile the plane, this is the same thing, but for irregular shapes that can also tile the plane, such as the polytan (45-45-90 triangle), polydrafter (30-60-90 triangle), and polypon (30-30-120 triangle), there is a differences. Some, such as the polykite (60-90-120-90 Quadrilateral) and polycairo are usually only based on their defining grid. Edge connected vs vertex connected: Most polyforms are connected edge to edge, but regular grid shapes can also be considered connected by vertexes. This includes squares and triangles. Hexagons cannot do this, since there are only three hexagons around a vertex, so they will all be connected by an edge. Variant enumerations The standard enumeration of polyforms is called "double-sided" meaning they can be flipped over, and rotated. This means that the mirror image of asymmetric forms are equivalent, and only counted once. "Single-sided" means that they cannot be flipped, so mirror images are distinct forms, but rotations are still counted as one. The difference between double-sided and single-sided of a given size indicate how many are asymmetric. "Fixed" enumeration counts all possible unique orientations, rotation and mirror images, as distinct. This mostly applies to polyforms that are grid based, and orientations relative to the grid. "Rooted" polyforms have one of the base shaped specially designated as the root. They can be double-sided, single-sided, or fixed. "Holeless" excludes any shapes that enclose an empty space, while "Holed" only counts polyforms with holes. "Path" (also called "Strip") polyforms are simple lines, each shape only touches at most two other shapes, with two ends touching only one other. Only the double-sided versions will be listed explicitly, as the other versions can be derived from them. Category:Types of polyforms